The eccentricity of the hyperbola x2a2-y2b2 - -1 is given by

For given hyperbola nbsp; x2a2 y2b2= 1i.e. #160;y2b2 x2a2=1eccentricity #160;e=b2+a2b2 #160; #160; #160; #160; #160; #160; #160; #160; # ...

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Standard XII

Mathematics

Conjugate Hyperbola

The eccentric...

Question

The eccentricity of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}}$ = –1 is given by __________________.

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Solution

For given hyperbola
$\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = - 1 i . e . \frac{y^{2}}{b^{2}} - \frac{x^{2}}{a^{2}} = 1 eccentricity e = \sqrt{\frac{b^{2} + a^{2}}{b^{2}}} i . e . e^{2} = \frac{b^{2} + a^{2}}{b^{2}} i . e . b^{2} e^{2} = b^{2} + a^{2} i . e . a^{2} = b^{2} (c^{2} - 1)$

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If the eccentricities of the hyperbolas $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 a n d \frac{y^{2}}{b^{2}} - \frac{x^{2}}{a^{2}} = 1$ be e and $e_{1}$ , then $\frac{1}{e^{2}} + \frac{1}{e_{1}^{2}} =$

Q. The eccentricity of the hyperbola

\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}}

=1 which passes through the points(3, 0) and

(3 \sqrt{2}, 2)

, is ________________.

Q. If the distance between the foci and the distance between the directries of the hyperbola

\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1

are in the ratio 3:2, then its eccentricity is __________________.

Q. The eccentricity of an ellipse

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1,

whose latus rectum is half of its major axis, is .

Q. Find the condition for the following set of curves to intersect orthogonally:

(i)

\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 and x y = c^{2}

(ii)

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 and \frac{x^{2}}{A^{2}} - \frac{y^{2}}{B^{2}} = 1

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The eccentricity of the hyperbola x2a2-y2b2 = –1 is given by __________________.

For given hyperbola x2a2-y2b2=-1i.e. y2b2-x2a2=1eccentricity e=b2+a2b2 i.e. e2=b2+a2b2 i.e. b2 e2=b2+a2 i.e. a2=b2c2-1

The eccentricity of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}}$ = –1 is given by __________________.

For given hyperbola
$\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = - 1 i . e . \frac{y^{2}}{b^{2}} - \frac{x^{2}}{a^{2}} = 1 eccentricity e = \sqrt{\frac{b^{2} + a^{2}}{b^{2}}} i . e . e^{2} = \frac{b^{2} + a^{2}}{b^{2}} i . e . b^{2} e^{2} = b^{2} + a^{2} i . e . a^{2} = b^{2} (c^{2} - 1)$